The equation `y=4 + 2 sin (6t - 3x)` represents a wave motion with

Options

A. amplitude 6 units
B. amplitude 2 units
C. wave speed 2 units
D. wave speed `1//2` units

Answer

To analyze the wave motion represented by the equation \( y = 4 + 2 \sin(6t - 3x) \), we will extract the relevant parameters step by step.

### Step 1: Identify the General Form of the Wave Equation
The general form of a wave equation is given by:
\[ y = B + A \sin(\omega t - kx) \]
where:
- \( B \) is the vertical shift (also called the equilibrium position),
- \( A \) is the amplitude of the wave,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Compare the Given Equation with the General Form
The given equation is:
\[ y = 4 + 2 \sin(6t - 3x) \]
From this, we can identify:
- \( B = 4 \)
- \( A = 2 \)
- \( \omega = 6 \)
- \( k = 3 \)

### Step 3: Determine the Amplitude
The amplitude \( A \) of the wave is the coefficient of the sine function:
\[ \text{Amplitude} = A = 2 \]

### Step 4: Calculate the Wave Speed
The wave speed \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values we found:
\[ v = \frac{6}{3} = 2 \]

### Step 5: Summary of Results
- The amplitude of the wave is \( 2 \) units.
- The wave speed is \( 2 \) units.

### Final Answer
The equation \( y = 4 + 2 \sin(6t - 3x) \) represents a wave motion with:
- Amplitude: \( 2 \) units
- Wave Speed: \( 2 \) units

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